Multi-utility representations of incomplete preferences induced by set-valued risk measures

We establish a variety of numerical representations of preference relations induced by set-valued risk measures. Because of the general incompleteness of such preferences, we have to deal with multi-utility representations. We look for representations that are both parsimonious (the family of representing functionals is indexed by a tractable set of parameters) and well behaved (the representing functionals satisfy nice regularity properties with respect to the structure of the underlying space of alternatives). The key to our results is a general dual representation of set-valued risk measures that unifies the existing dual representations in the literature and highlights their link with duality results for scalar risk measures.


Introduction
This note is concerned with the numerical representation of preference relations induced by a special class of set-valued maps. Recall that a preference (relation) over the elements of a set L is a reflexive and transitive binary relation on L. A preference is said to be complete if any two elements x, y ∈ L are comparable in the sense that it is always possible to determine whether x is preferred to y or viceversa. Following the terminology of Dubra et al. [14], a family U of maps u : L → [−∞, ∞] is a multi-utility representation of a preference if for all x, y ∈ L we have x y ⇐⇒ u(x) ≥ u(y) for every u ∈ U .
In words, a multi-utility representation provides a numerical representation for the given preference relation via a family of "utility functionals". In view of their greater tractability, multi-utility representations play a fundamental role in applications. A standard problem in this context is to find representations that are at the same time parsimonious (the family of representing functionals is indexed by a small set of parameters) and well behaved (the representing functionals satisfy nice regularity properties with respect to the structure of the underlying set). This is especially important for incomplete preferences, which cannot be represented by a unique functional.
The goal of this note is to establish numerical representations of preference relations induced by a special class of set-valued maps that have been the subject of intense research in the recent mathematical finance literature. To introduce the underlying economic problem, consider an economic agent who is confronted with the problem of ranking a number of different alternatives represented by the elements of a set L. The agent has specified a target set of acceptable or attractive alternatives A ⊆ L. We assume that, if an alternative is not acceptable, it can be made acceptable upon implementation of a suitable admissible action. We represent the results of admissible actions by the elements of a set M ⊆ L and assume that a given alternative x ∈ L can be transformed through a given m ∈ M into the new alternative x + m. The objective of the agent is then to identify, for each alternative, all the admissible actions that can be implemented to move said alternative inside the target set by way of translations. This naturally leads to the set-valued map R : L ⇒ M defined by The map R can be seen as a generalization of the set-valued risk measures studied by Jouini et al. [32], Kulikov [35], Hamel and Heyde [28], Hamel et al. [29], and Molchanov and Cascos [37] in the context of markets with transaction costs; by Haier et al. [27] in the context of intragroup transfers; by Feinstein et al. [22], Armenti et al. [3], and Ararat and Rudloff [1] in the context of systemic risk. We refer to these contributions for a discussion about the financial interpretation of set-valued risk measures in the respective fields of application and to Section 5 for some concrete examples in the context of multi-currency markets with transaction costs and systemic risk.
The set-valued map R defined above induces a preference relation on L by setting According to this preference, the agent prefers x to y if every admissible action through which we can move y into the target set will also allow us to transport x there. In other terms, x is preferred to y if it is easier to make x acceptable compared to y. The goal of this note is to establish numerical representations of the preference R . Since this preference, as shown below, is not complete in general, we have to deal with multi-utility representations. In particular, we look for representations consisting of (semi)continuous utility functionals. We achieve this by establishing suitable (dual) representations of the set-valued map R.
Our results provide a unifying perspective on the existing dual representations of set-valued risk measures and on the corresponding multi-utility representations, which, to be best of our knowledge, have never been explicitly investigated in the literature. We illustrate the advantages of such a unifying approach by discussing applications to multi-currency markets with transaction costs and systemic risk. In addition, we highlight where our strategy to establishing dual representations differs from the standard arguments used in the literature. The note is structured as follows. The necessary mathematical background is collected in Section 2. The standing assumptions on the space of alternatives and the main properties of the set-valued map under investigation are presented in Section 3. The main results on dual and multiutility representations are established in Section 4 and are applied to a number of concrete situations in Section 5.

Mathematical background
In this section we collect the necessary mathematical background and fix the notation and terminology used throughout the paper. We refer to Rockafellar [40] and Zȃlinescu [43] for a thorough presentation of duality for topological vector spaces. Moreover, we refer to Aubin and Ekeland [4] for a variety of results on support functions and barrier cones.
Let L be a real locally convex Hausdorff topological vector space. The topological dual of L is denoted by L ′ . Any linear subspace M ⊆ L is canonically equipped with the relative topology inherited from L. The corresponding dual space is denoted by M ′ . For every set A ⊆ L we denote by int(A) and cl(A) the interior and the closure of A, respectively. We say that A is convex if λA The effective domain of σ A is called the barrier cone of A and is denoted by It follows from the Hahn-Banach Theorem that, if A is closed and convex, then it can be represented as the intersection of all the halfspaces containing it or equivalently Finally, if A + K ⊆ A for some cone K ⊆ L, then bar(A) ⊆ K + .

The setting
Throughout the remainder of the note, we assume that L is a real locally convex Hausdorff topological vector space. We also fix a closed convex cone K ⊆ L satisfying K − K = L and consider the induced partial order defined by The above partial order is meant to capture an "objective" preference relation shared by all agents. This is akin to the "better for sure" preference in Drapeau and Kupper [13]. The next proposition collects a number of basic properties of the set-valued map R and its associated preference R . The properties of R are aligned with those discussed in Hamel and Heyde [28] and Hamel et al. [29].
Proposition 3.2. (i) R is monotone with respect to K, i.e. for all x, y ∈ L x K y =⇒ x R y.
(vi) R(x) is convex and closed for every x ∈ L.
Proof. To establish (i), assume that x K y for x, y ∈ L. For every m ∈ R(y) we have This shows that m ∈ R(x) as well, so that x R y. To establish (ii), take λ ∈ [0, 1] and assume that x R y. For every m ∈ R(y) we have y + m ∈ A and, hence, x + m ∈ A. This yields showing that m ∈ R(λx + (1 − λ)y). In sum, λx + (1 − λ)y R y. To see that properties (iii) to (vi) hold, it suffices to recall that R(x) = M ∩ (A − x) for every x ∈ L. Finally, to establish (vii), assume that R(x) = M for some x ∈ L. Take any y ∈ L and assume that R(y) is nonempty so that y + m ∈ A for some m ∈ M . For all n ∈ M and λ ∈ (0, 1] we have by convexity. Hence, letting λ → 0, we obtain y +n ∈ A by closedness. Since n was arbitrary, we infer that R(y) = M . This contradicts assumption (A3), showing that R(x) = M must hold for every x ∈ L.
Remark 3.3. (i) If M is spanned by a single element, then R is complete. Indeed, in this case, we can always assume that M is spanned by a nonzero element m ∈ M ∩ K by our standing assumption. Then, for every x ∈ L such that R(x) = ∅ we see that for a suitable λ x ∈ R. This shows that R is complete.
(ii) In general, the preference R is not complete when M is spanned by more than one element. For instance, let L = R 3 and assume that K = A = R 3 + and M = R 2 × {0}. For x = 0 and y = (1, −1, 0) we respectively have Clearly, neither x R y nor y R x holds, showing that R is not complete. (iii) Sometimes the preference R is complete even if M is spanned by more than one element. For instance, let L = R 3 and assume that This shows that R is complete.

Multi-utility representations
In this section we establish a variety of multi-utility representations of the preference induced by R, which are derived from suitable representations of the sets R(x). As highlighted below, both representations have a strong link with (scalar) risk measures and their dual representations. We refer to the appendix for the necessary mathematical background and notation.
The first multi-utility representation is based on the following scalarizations of R. Here, we set Moreover, we define a map u π : L → [−∞, ∞] by setting The functionals ρ π are examples of the risk measures introduced in Föllmer and Schied [23] and generalized in Frittelli and Scandolo [25]. We refer to Farkas et al. [19,20] for a thorough investigation of such functionals at our level of generality. The next proposition features some of their standard properties, which follow immediately from Proposition 3.2. Since the announced multi-utility representation will be expressed in terms of the negatives of the functionals ρ π , the proposition is stated in terms of the utility functionals u π .
Proposition 4.2. For every π ∈ K + M the functional u π satisfies the following properties: (i) u π is translative along M , i.e. for all x ∈ L and m ∈ M u π (x + m) = u π (x) + π(m).
(ii) u π is nondecreasing with respect to K , i.e. for all x, y ∈ L x K y =⇒ u π (x) ≥ u π (y).
(iii) u π is concave, i.e. for all x, y ∈ L and λ ∈ [0, 1] Remark 4.3. Note that, unless M is spanned by one element, the closedness of the set A is not sufficient to ensure that the functionals ρ π are lower semicontinuous; see Example 1 in Farkas et al. [20]. We refer to Hamel et al. [30] for a discussion on general sufficient conditions ensuring the lower semicontinuity of scalarizations of set-valued maps and to Farkas et al. [20] and Baes et al. [6] for a variety of sufficient conditions in a risk measure setting.
The first multi-utility representation of the preference induced by R rests on the intimate link between the risk measures ρ π and the support functions corresponding to R.

Lemma 4.4.
For every x ∈ L the set R(x) can be represented as Proof. The result is clear if R(x) = ∅. Otherwise, recall that R(x) is closed and convex by Proposition 3.2 and observe that ρ π (x) = σ R(x) (π) for every π ∈ M ′ . We can apply the dual representation (2.1) in the context of the space M to obtain As R(x) + K ∩ M ⊆ R(x) again by Proposition 3.2, we conclude by noting that the barrier cone of R(x) must be contained in K + M .
Theorem 4.5. The preference R can be represented by the multi-utility family Proof. We rely on Lemma 4.4. Take any x, y ∈ L. If x R y, then R(x) ⊇ R(y) and This yields x R y and concludes the proof.
Remark 4.6. The simple representation in Lemma 4.4 shows that the set-valued map R is completely characterized by the family of functionals ρ π . In the context of risk measures, one could say that a setvalued risk measure is completely characterized by the corresponding family of scalar risk measures. This corresponds to the "setification" formula in Section 4.2 in Hamel et al. [30].
We aim to improve the above representation in a twofold way. First, we want to find a multi-utility representation consisting of a smaller number of representing functionals. This is important to ensure a more parsimonious, hence tractable, representation. Second, we want to establish a multi-utility representation consisting of (semi)continuous representing functionals. This is important in applications, e.g. in optimization problems where the preference appears in the optimization domain.
The second multi-utility representation will be expressed in terms of the following utility functionals. Here, for any functional π ∈ M ′ we denote by ext(π) the set of all linear continuous extensions of π to the whole space L, i.e. ext(π) := {ψ ∈ L ′ : ψ(m) = π(m), ∀m ∈ M }.
(If A is a cone, then σ A = 0 on bar(A) and the above maps simplify accordingly).
The functionals ρ * π are inspired by the dual representation of the risk measures ρ π , see e.g. Frittelli and Scandolo [25] or Farkas et al. [20]. The precise link is shown in Proposition 4.14 below. For the time being, we are interested in highlighting some properties of the functionals ρ * π , or equivalently u * π , and proceeding to our desired multi-utility representation.
Proposition 4.8. For every π ∈ K + M the functional u * π satisfies the following properties: Proof. Translativity follows from the definition of ρ * π . Being a supremum of affine maps, it is clear that ρ * π is convex and lower semicontinuous. To show monotonicity, it suffices to observe that bar(A) ⊆ K + by (A1) and therefore To streamline the proof of the announced multi-utility representation, we start with the following lemma. We denote by ker(π) the kernel of π ∈ M ′ , i.e. ker(π) := {m ∈ M : π(m) = 0}.
In the sequel, we will repeatedly use the fact that ker(π) has codimension 1 in M (provided π is nonzero).
Since x ∈ cl(A + ker(π ψ )) by our assumption, we can use (2.1) again to get where the last equality holds because ψ ∈ ker(π ψ ) ⊥ . This concludes the proof.
The next lemma records a representation of the map R that will immediately yield our desired multi-utility representation with (upper) semicontinuous functionals.
Lemma 4.10 (Dual representation of R). For every x ∈ L the set R(x) can be represented as (If A is a cone, then σ A = 0 on bar(A) and the representation simplifies accordingly).
Proof. Fix x ∈ L. It follows from the representation in (2.1) and Lemma 4.9 that To establish the desired representation of R(x) it then suffices to show that the set ker(π) ⊥ in the righthand side of (4.2) can be replaced by ext(π). To this effect, let m ∈ M satisfy π(m) ≥ σ A (ψ) − ψ(x) for all nonzero π ∈ K + M and ψ ∈ ext(π). Moreover, take an arbitrary nonzero π ∈ K + M and an arbitrary ψ ∈ ker(π) ⊥ . To conclude the proof, we have to show that This is clear if ψ / ∈ bar(A) or ψ ∈ ext(π). Hence, assume that ψ ∈ bar(A) \ ext(π). Note that, since π is nonzero and K − K = L, we find n ∈ K M such that π(n) > 0. Since bar(A) ⊆ K + , two situations are possible. On the one hand, if ψ(n) > 0, then ψ belongs to ext(π) up to a strictly-positive multiple and therefore (4.3) holds. On the other hand, if ψ(n) = 0, then we must have ψ ∈ M ⊥ . To deal with this case, note first that we always find a nonzero π * ∈ K + M satisfying ext(π * ) ∩ bar(A) = ∅, for otherwise every functional in bar(A) ∩ ker(π * ) ⊥ would annihilate the entire M and it would follow from (2.1) and (4.2) that R(y) = M for every y ∈ A, which is against Proposition 3.2. Now, take ϕ ∈ ext(π * ) ∩ bar(A) and set ϕ k = ϕ + kψ ∈ ext(π * ) for each k ∈ N. It follows that This implies that ψ(m) = 0 ≥ σ A (ψ) − ψ(x) must hold, establishing (4.3).
Theorem 4.11. The preference R can be represented by the multi-utility family Proof. Note that ρ * π (x) = −∞ for every x ∈ L whenever ext(π) ∩ bar(A) = ∅ for some π ∈ K + M . Hence, the desired assertion follows immediately from Lemma 4.10; see also the proof of Theorem 4.5. The next proposition shows the link between the two multi-utility representations we have established. In a sense made precise below, the representation U * can be seen as the regularization of U by means of (upper) semicontinuous hulls. Before we show this, it is useful to single out the following dual representation of the augmented acceptance set, which should be compared with Theorem 1 in Farkas et al. [20]. {x ∈ L : ψ(x) ≥ σ A (ψ)}.
Proof. In view of (2.1) and Lemma 4.9, the assertion is equivalent to We only need to show the inclusion "⊇". To this end, we mimic the argument in the proof of Lemma 4.10. Let x ∈ L belong to the right-hand side above and take ψ ∈ ker(π) ⊥ . We have to show that This is clear if ψ / ∈ bar(A) or ψ ∈ ext(π). Hence, assume that ψ ∈ bar(A) \ ext(π). Note that, since π is nonzero and K − K = L, we find n ∈ K M such that π(n) > 0. Since bar(A) ⊆ K + , two situations are possible. On the one hand, if ψ(n) > 0, then ψ belongs to ext(π) up to a strictly-positive multiple and therefore (4.4) holds. On the other hand, if ψ(n) = 0, then we must have ψ ∈ M ⊥ . In this case, take any functional ϕ ∈ ext(π) ∩ bar(A) and set ϕ k = kψ + ϕ ∈ ext(π) for every k ∈ N. Then, for every k ∈ N. Letting k → ∞ yields (4.4) and concludes the proof.  (ii) u * π = usc(u π ). Proof. Fix a nonzero π ∈ K + M such that ext(π) ∩ bar(A) = ∅. Clearly, we only need to show (i). To this effect, recall that ρ * π is lower semicontinuous and note that it is dominated by ρ π . Indeed, for every x ∈ L and for every m ∈ M such that x + m ∈ A sup ψ∈ext(π) showing that ρ * π (x) ≤ ρ π (x). Now, take a lower semicontinuous map f : L → [−∞, ∞] such that f ≤ ρ π . We claim that f ≤ ρ * π as well. To show this, suppose to the contrary that f (x) > ρ * π (x) for some x ∈ L. Note that ρ * π (x) = inf{λ ∈ R : x + λm ∈ cl(A + ker(π))} by Lemma 4.13, where m ∈ M is any element satisfying π(m) = 1 (which exists because π is nonzero). As a result, we must have f (x) > λ for some λ ∈ R such that x + λm ∈ cl(A + ker(π)). Hence, there exist two nets (x α ) ⊆ A and (m α ) ⊆ ker(π) such that x α + m α → x + λm. Since {f > λ} is open by lower semicontinuity, it eventually follows from the translativity of ρ π that Since this is impossible, we infer that f ≤ ρ * π must hold, concluding the proof.
Remark 4.15. (i) The preceding proposition shows that the dual representation in Lemma 4.10 and, hence, the multi-utility representation in Theorem 4.11 can be equivalently stated in terms of the semicontinuous hulls of the functionals ρ π and u π , respectively. This should be compared with the representation in Lemma 5.1 in Hamel and Heyde [28].
(ii) The preceding proposition also suggests the following alternative path to establishing Lemma 4.10: (1) Start with the representation in Lemma 4.4.
(2) Show that the functionals ρ π there can be replaced by their lower semicontinuous hulls lsc(ρ π ).
(3) Show that we can discard from the representation all the functionals π ∈ K + M \ {0} such that lsc(ρ π ) is not proper or, equivalently, ext(π) ∩ bar(A) = ∅. (4) Use Proposition 4.14 to replace the functionals lsc(ρ π ) with the more explicit functionals ρ * π . The advantage of the strategy pursued in the proof of Lemma 4.10 is that it avoids passing through semicontinuous hulls and the analysis of their properness.
The representing functionals belonging to the multi-utility representation in Theorem 4.11 are, by definition, upper semicontinuous. As a final step, we want to find conditions ensuring a multi-utility representation consisting of continuous functionals only. To achieve this, we exploit the link between the functionals ρ π and their regularizations ρ * π established in Proposition 4.14.
Proof. First of all, we claim that ρ π (x) > −∞ for every x ∈ L. To see this, take any functional ψ ∈ ext(π) ∩ bar(A) and note that for every As a result, ρ π is finite valued. Note that, by definition, ρ π is bounded above on A by 0. Since A has nonempty interior and ρ π is convex, we infer from Theorem 8 in Rockafellar [40] that ρ π is continuous. The last statement is a direct consequence of Proposition 4.14.
The following multi-utility representation with continuous utility functionals is a direct consequence of Theorem 4.11 and Lemma 4.16.
Theorem 4.17. Assume that int(A) = ∅ and that ρ π (x) < ∞ for all π ∈ K + M \{0} with ext(π)∩bar(A) = ∅ and x ∈ L. Then, the preference R can be represented by the multi-utility family In addition, every element of U * * is finite valued and continuous.
We conclude by showing a number of sufficient conditions for the finiteness assumption in Lemma 4.16 to hold. This should be compared with the results in Section 3 in Farkas et al. [20]. The recession cone of A is denoted by rec(A) := {x ∈ L : x + y ∈ A, ∀y ∈ A}.  Proof. The desired assertion clearly holds under (i). Since K ⊆ rec(A) by assumption (A1), we see that qint(K) ⊆ qint(rec(A)). Hence, it suffices to establish that (iii) implies the desired assertion. So, assume that (iii) holds and take m ∈ M ∩ qint(rec(A)). If ρ π (x) = ∞ for some x ∈ L, then we must have (x + M ) ∩ A = ∅. It follows from a standard separation result, see e.g. Theorem 1.1.3 in Zȃlinescu [43], that we find a nonzero functional ψ ∈ L ′ satisfying ψ(x + λm) ≤ σ A (ψ) for every λ ∈ R. This is only possible if ψ(m) = 0, which cannot hold because ψ ∈ bar(A) ⊆ (rec(A)) + . As a result, we must have ρ π (x) < ∞ for every x ∈ L.

Applications
In this final section we specify the general dual representation of R to a number of concrete situations. The explicit formulation of the corresponding multi-utility representation can be easily derived as in Theorem 4.11 and Theorem 4.17. Throughout the section we consider a probability space (Ω, F, P) and fix an index d ∈ N. For every p ∈ [0, ∞] and every Borel measurable set S ⊆ R d we denote by L p (S) the set of all equivalence classes with respect to almost-sure equality of d-dimensional random vectors X = (X 1 , . . . , X d ) : Ω → R d with p-integrable components such that P[X ∈ S] = 1. As usual, we never explicitly distinguish between an equivalence class in L p (S) and any of its representative elements. We treat R d as a linear subspace of L p (R d ). For all vectors a, b ∈ R d we set The expectation with respect to P is simply denoted by E. For every p ∈ [1, ∞] the space L p (R d ) can be naturally paired with L q (R d ) for q = p p−1 via the bilinear form Here, we adopt the usual conventions 1 0 := ∞ and ∞ ∞ := 1. Finally, for every random vector X ∈ L 1 (R d ) we use the compact notation E[X] := (E[X 1 ], . . . , E[X d ]).

Set-valued risk measures in a multi-currency setting
We consider a financial market where d different currencies are traded. Every element of L 1 (R d ) is interpreted as a vector of capital positions expressed in our different currencies at some future point in time. For a pre-specified acceptance set A ⊆ L 1 (R d ) we look for the currency portfolios that have to be set up at the initial time to ensure acceptability.

The static case
As a first step, we consider a one-period market with dates 0 and 1. In this setting, we focus on the currency portfolios that we have to build at time 0 in order to ensure acceptability of currency positions at time 1. This naturally leads to defining the set-valued map R : Assumption 5.1. In this subsection we work under the following assumptions: (1) A is norm closed, convex, and satisfies A + L 1 (R d + ) ⊆ A.
We derive the following representation by applying our general results to This result should be compared with the dual representation established in Jouini et al. [32], Kulikov [35], and Hamel and Heyde [28].
Proposition 5.2. For every X ∈ L 1 (R d ) the set R(X) can be represented as In addition, if A is a cone, then we can simplify the above representation using that Proof. Note that K + M can be identified with R d + and that bar(A) is contained in L ∞ (R d + ) by assumption (1). Since, for all w ∈ R d and Z ∈ L ∞ (R d ), the random vector Z (viewed as a functional on L 1 (R d )) is an extension of w (viewed as a functional on R d ) precisely when E[Z] = w, the desired representation follows immediately from Lemma 4.10. Example 5.4 (Multidimensional Expected Shortfall). For every X ∈ L 1 (R) and every α ∈ (0, 1) we denote by ES α (X) the Expected Shortfall of X at level α, i.e.
where q X is any quantile function of X. The multi-dimensional acceptance set based on Expected Shortfall introduced in Hamel et al. [31] is given by for a fixed α = (α 1 , . . . , α d ) ∈ (0, 1) d . Note that assumptions (1) and (2) hold. In particular, we have for every w ∈ R d + (where w α is understood component by component). This follows from the standard dual representation of Expected Shortfall; see Theorem 4.52 in Föllmer and Schied [24]. As a result, the dual representation in Proposition 5.2 reads for every random vector X ∈ L 1 (R d ).

The dynamic case
As a next step, we consider a multi-period financial market with dates t = 0, . . . , T and information structure represented by a filtration (F t ) satisfying F 0 = {∅, Ω} and F T = F. In this setting, currency portfolios can be rebalanced through time. A (random) portfolio at time t ∈ {0, . . . , T } is represented by an F t -measurable random vector in L 0 (R d ). We denote by C t the set of F t -measurable portfolios that can be converted into portfolios with nonnegative components by trading at time t. This means that, for all F t -measurable portfolios m t and n t , we can exchange m t for n t at time t provided that m t − n t ∈ C t . The sets C t are meant to capture potential transaction costs. A flow of portfolios is represented by an adapted process (m t ). More precisely, for every date t ∈ {0, . . . , T − 1}, the portfolio m t is set up at time t and held until time t + 1. The portfolio flows belonging to the set are said to be admissible. The admissibility condition is a direct extension of the standard self-financing property in frictionless markets.
We look for all the initial portfolios that can be rebalanced in an admissible way until the terminal date in order to ensure acceptability. This leads to the set-valued map R : In words, the above set consists of all the initial portfolios that give rise, after a convenient exchange at date 0, to an admissible rebalancing process making the outstanding currency position acceptable after a final portfolio adjustment at time T . This setting can be embedded in our framework because we can equivalently write Assumption 5.5. In this subsection we work under the following assumptions: (1) A is norm closed, convex, and satisfies A + L 1 (R d + ) ⊆ A.
We derive the following representation by applying our general results to For convenience, we also set For later use note that The next result should be compared with the dual representation established in Hamel et al. [29] in the special setting of Example 5.8.
Proposition 5.6. For every X ∈ L 1 (R d ) the set R(X) can be represented as where we have set for every Z ∈ L ∞ (R d ) In addition, if A is a cone, the above representation can be simplified by using that Moreover, if C 0 is a cone, then Similarly, if C t is a cone for every t ∈ {1, . . . , T }, then Proof. The assertion follows from Proposition 5.
Remark 5.7. Note that, as in the static case, we have M ∩ qint(K) = ∅. This can be used to ensure multi-utility representations with continuous representing functionals; see Proposition 4.18.
Example 5.8 (Superreplication under proportional transaction costs). We adopt the discrete version of the model by Kabanov [33]. For every t ∈ {0, . . . , T } we say that a set-valued map S : for every open set U ⊂ R d . In this case, we denote by L 0 (S) the set of all random vectors X ∈ L 0 (R d ) such that P[X ∈ S] = 1. This set is always nonempty if S has closed values; see Corollary 14.6 in Rockafellar and Wets [41]. Now, let K t : Ω ⇒ R d be an F t -measurable set-valued map such that K t (ω) is a polyhedral convex cone (hence K t (ω) is closed) containing R d + for every ω ∈ Ω and set Moreover, we consider the worst-case acceptance set Assumptions (1) and (2) are easily seen to be satisfied. Moreover, A as well as each of the sets C t is a cone. As proved in Theorem 2.1 in Schachermayer [42], assumption (3) always holds under the so-called "robust no-arbitrage" condition. Finally, as 0 ∈ R(0), assumption (4) holds if and only if R d is not entirely contained in T t=0 C t . Note also that A + = L ∞ (R d + for every X ∈ L 1 (R d ). The dual elements Z in the above representation can be linked to consistent pricing systems, see e.g. Schachermayer [42]. To see this, note that, for every t ∈ {0, . . . , T }, the set-valued map K + t : Ω ⇒ R d defined by K + t (ω) = K t (ω) + is F t -measurable, see e.g. Exercise 14.12 in Rockafellar and Wets [41], and such that by measurable selection, see the argument in the proof of Theorem 1.7 in Schachermayer [42]. As a result, every dual element Z in the above dual representation satisfies This shows that the d-dimensional adapted process (E[Z|F t ]), where the conditional expectations are taken componentwise, satisfies E[Z|F T ] = Z and E[Z|F t ] ∈ L 0 (K + t ) for every t ∈ {0, . . . , T } and thus qualifies as a consistent pricing system. In other words, the above dual elements Z can be viewed as the terminal values of consistent pricing systems. Remark 5.9. (i) It is worth noting that our approach provides a different path, compared to the strategy pursued in Schachermayer [42], to establish the existence of consistent pricing systems under the robust no-arbitrage assumption (admitting the closedness of the reference target set). Moreover, by rewriting the above dual representation in terms of consistent pricing systems, we recover the (localization to L 1 (R d ) of the) superreplication theorem by Schachermayer [42].
(ii) The above dual representation was also obtained in Hamel et al. [29]. Differently from that paper, we have not derived it from the superreplication theorem in Schachermayer [42] but from a direct application of our general results.

Systemic set-valued risk measures based on acceptance sets
We consider a single-period economy with dates 0 and 1 and a financial system consisting of d entities. for every random vector X ∈ L ∞ (R d ).